A pr 1 99 9 Are the Dirac particles of the Standard Model dynamically confined states in a higher - dimensional flat space ? 1 Ronald

Some time ago Rubakov and Shaposhnikov suggested that elementary particles might be excitations trapped on a soliton in a flat higher dimensional space. They gave as an example φ 4 theory in five dimensions with a bosonic excitation on a domain wall φ cl in the fifth dimension. They also trapped a chiral Dirac particle on the domain wall with the interaction lagrangian gψφψ. The field equation for the Dirac field was −i 4 µ=0 γ µ ∂ µ ψ−gφψ = 0. We show that a Dirac equation in eight flat dimensions with −gφ replaced by a harmonic-oscillator " potential " in the four higher dimensions (plus a large constant M 0) generates bound states in approximate SU (4) × SU (2) representations. These representations have corresponding " orbital " times " spin " quantum numbers and bear some resemblance to the quarks and leptons of the Standard Model. Soliton-theory suggests that the harmonic-oscillator potential should rise only a finite amount. This will then limit the number of generations in a natural way. It will also mean that particles with sufficient energy might escape the well altogether and propagate freely in the higher dimensions. It may be worthwhile to search for a soliton (instanton?) which can confine Dirac particles in the manner of the harmonic-oscillator potential.